The existence of a suitable enlargement \(\hat{\mathcal {K}}\)  such that \(\mathcal {K} \subset \text{Int\ } \hat{\mathcal {K}}\)  is guaranteed by smoothness. In particular, following the analogous setup for ODEs in [1]}, [2]}, \(\hat{\mathcal {K}}\)  can be chosen so that \(S\)  is defined/extended over \(\hat{\mathcal {K}}\)  as a \(C^r-\) smooth graph
\(\hat{S} = \left\lbrace (x,\phi _0(x)) : x \in \hat{\mathcal {K}} \right\rbrace  ,\) 
